# Questions tagged [line-bundles]

A continuously varying family of one-dimensional vector spaces over a topological space. A related tag is the vector-bundles tag.

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### The dual of the space of continuous sections in a vector bundle

If $X$ is a compact Hausdorff space, one may view the space of complex, continuous functions on it as the space of continuous sections in the trivial Hermitian bundle $X \times \mathbb C$. By the ...

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**1**answer

609 views

### Embedding abelian varieties into projective spaces of small dimension

Given a (complex) abelian variety $A$ of a fixed dimension $g$, let $d(A)$ be the dimension of the smallest complex projective space it embeds into.
Is $d(A)$ uniform over all abelian varieties of a ...

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**1**answer

138 views

### The existence of the extension of a non-trivial line bundle

In Three Dimensional Gravity Revisted, Witten studied the Abelian Chern-Simons theory in three dimensions.
Let $W$ be a three dimensional manifold. Let $\mathcal{L}$ be a non-trivial line-bundle over ...

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votes

**1**answer

89 views

### Pull-back of polarization

Let $(X, L)$ and $(Y, M)$ be two polarized abelian varieties .
According to Birkenhake C. and Lange H. in Complex Abelian Varieties a homomorphism of polarized abelian varieties $f:(Y, M)\...

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146 views

### Does there exist a preferred trivialization of a trivial line bundle?

Let $L\to M$ be a topologically trivial complex Hermitian line bundle (over a manifold of dimension three, if this is of any importance). I assume that $L$ admits a trivialization, however, I do not ...

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362 views

### Classification of line bundles by second cohomology of a manifold

In the book Loop spaces, Characteristic classes and geometric quantization by Brylinski I see following result when trying to motivate geometric description of $H^3(M,\mathbb{Z})$.
$H^2(M,\mathbb{Z}...

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votes

**2**answers

251 views

### Is a torsion free sheave of rank one on a reducible curve the pushforward of a line bundle on a normalization?

Let $X$ be a nodal curve, possibly reducible. Then can any torsion free sheaf of rank one on $X$ be expressed as $\pi_*(L)$, where $L$ is a line bundle on a partial normalization of $X$? This looks ...

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### $H$ very ample, $f$ finite, is there uniform $C=C(\mathrm{deg}(f))$ for $C f^* H$ very ample?

Let $X$ and $Y$ be normal projective varieties over $\mathbb{C}$ of dimension $n$. Let $f: X \rightarrow Y$ be a finite morphism. Also, let $H$ be a very ample divisor on $Y$.
Is there a constant $C=...

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votes

**1**answer

142 views

### Do line bundles with enough sections on surfaces have generic divisors which are irreducible?

Let $L$ be a line bundle on a smooth connected complete complex algebraic surface $X$. Assume that $L$ has enough sections i.e. that $H^0(L,X)$ has dimension $> 1$. A nonzero section $s$ of $L$ ...

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191 views

### When is the space of holomorphic sections of the tensor product of two line bundles given by the span of the tensor product of the basis?

Let $S$ be a compact complex manifold and $L_1, L_2 \longrightarrow S$
be two holomorphic line bundles. Under what conditions (hopefully something that is easy to check) on $L_1$ and $L_2$ is the ...

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vote

**1**answer

142 views

### Reference request: $f^*D$ semi-ample, then $D$ semi-ample

I am looking for a suitable reference to put in a bibliography for the following fact:
Let $f: X \rightarrow Y$ be a surjective morphism between normal projective varieties. Let $D$ be a $\mathbb{Q}$-...

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vote

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110 views

### Torsion line bundle on hyperelliptic curves and Weierstrass points

Let $C$ be an hyperelliptic curve of genus $g$ and let $f:C\rightarrow
\mathbb{P}^1$ be the corresponding 2 to 1 covering
ramified in $2g+2$ points.
Let $L$ be a line bundle on $C$ such that either $...

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vote

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87 views

### Twisting a line bundle with the zero section

Let $X$ be a smooth projective curve and $L$ be an invertible sheaf on $X$. Denote by $\mathbb{L}$ the line bundle associated to $L$, $\pi:\mathbb{L} \to X$ the natural morphism and $0_\pi$ the zero ...

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**1**answer

372 views

### Isomorphism classes of line bundles with connections

Isomorphism classes of line bundles over a scheme $X$ are described by the Picard group $Pic(X)$. Now there is a paper that describes the moduli space of line bundles with connections. This paper is ...

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votes

**1**answer

153 views

### Curves contracted by a rational map

Let $D$ be a big but not nef divisor on a normal $\mathbb{Q}$-factorial projective variety. Assume that the section ring
$$R(D) = \bigoplus_{n\in\mathbb{N}}H^0(X,nD)$$
is finitely generated and ...

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### Determinant of the universal bundle

Let $M$ be the moduli space of semistable vector bundles of fixed determinant $L$ and rank $r$ over a smooth curve $X$. Assume that $gcd(r,deg(L))=1$. Let $\mathcal U$ be the universal bundle over $M\...

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47 views

### Connection on line bundle over general simplicial toric variety

In https://arxiv.org/pdf/hep-th/0005247.pdf, on page 60 and 61, it is mentioned that the connection of $\mathcal{O}(-n)$ over a (simplicial) toric variety of the form
$$
(\mathbb{C}^N \backslash U)/(\...

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410 views

### Picard group of toric varieties

I am trying to understand how to obtain the Picard group for general toric varieties. So far, I have been using information found in https://arxiv.org/pdf/1003.5217.pdf .
Here, a toric variety has ...

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105 views

### Holomorphic line bundles associated to multiple U(1) groups, defined over toric manifolds

The sections of the holomorphic line bundle $\mathcal{O}(n)$ are acted on by the covariant derivative
$$
d+nA,
$$
where $A$ is the connection on the $U(1)$ bundle to which $\mathcal{O}(n)$ is ...

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**2**answers

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### Square root of the determinant line

Let $\Sigma$ be a compact Riemann surface equipped with a spin structure (a square root of $\Omega^1_\Sigma$, denoted $\Omega^{1/2}_\Sigma$).
Let $\Gamma(\Omega^1_\Sigma)$ be the space of holomorphic ...

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47 views

### Minimal non-klt center of asymptotic linear system

Let $(X,\Delta)$ be a klt pair and $D $ a $Q $-Cartier divisor on $X $ such that the ring of sections of $D $ is finitely generated. Let $c$ be the log canonical threshold of the asymptotic linear ...

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291 views

### The kernel of a nef line bundle

Let $V$ be a complex projective variety and $L$ a nef line bundle on $V$ (i.e., $L$ is non-negative on every curve in $V$). Denote, as usual, $\deg_LX = c_1(L)^{\dim{X}}.[X]$ for $X$ a subvariety of $...

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**1**answer

122 views

### Anti-canonical divisorial contractions of weak Fano $3$-folds

Let $X$ be a smooth weak Fano but not Fano $3$-fold ($-K_X$ is nef and big but not ample). Then the anti-canonical morphism $\phi:X\rightarrow W$ (the morphsim induced by the linear system $|-mK_X|$ ...

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### Holomorphic line bundles on smooth points of a quotient

I am an amateur algebraic geometer, so maybe this question is trivial and if this is the case, then I apologize. This is a question that came up while working on something completely different.
...

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149 views

### In search for examples concerning pushforward of nef divisors and lc-trivial fibrations

My question is motivated by ideas around the moduli b-divisor of an lc-trivial fibration (see for instance the following paper by Ambro https://arxiv.org/pdf/math/0308143.pdf).
In such a setup, one ...

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votes

**2**answers

287 views

### Rational maps and Kodaira dimension

Let $\phi:X\dashrightarrow Y$ be a generically finite, dominant rational map between smooth projective varieties over $\mathbb{C}$.
Assume that $Y$ is of general type. May we conclude then that $X$ ...

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**1**answer

309 views

### Rationality of conic bundles

Let $\pi:X\rightarrow\mathbb{P}^2$ be a $3$-fold conic bundle, and let $\Delta\subset\mathbb{P}^2$ be its discriminant. Assume that both $X$ and $\Delta$ are smooth and that $deg(\Delta)\geq 6$.
Can ...

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### Nef line bundles on complex analytic spaces

Let $L$ be a line bundle over a compact complex manifold $X$ with a Hermitian metric $\omega$: $L$ is said numerically effective (nef, for short) if for any $\epsilon>0$ there exists a smooth ...

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### Lefschetz type theorems for big and nef divisors

Let $X$ be a smooth projective variety, and $D\subset X$ a smooth nef and big divisor. Assume that the restriction map $Pic(X)\rightarrow Pic(D)$ is an isomorphism over $\mathbb{Q}$.
Under which ...

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**1**answer

161 views

### Divisor class group of quartic surfaces

Let $X\subset\mathbb{P}^3$ be a normal quartic surface with divisor class group $Cl(X)\cong\mathbb{Z}[H]$ generated by the hyperplane section.
What can we say about the singularities of $X$?

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### Do line bundles which divide the canonical bundle lift

Let $S $ be the spectrum of a complete dvr with algebraically closed residue field. Let $\eta$ be its generic point and let $s$ be its closed point with $k(s)$ of positive characteristic. Let $X\to S$ ...

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299 views

### Lifting line bundles

Let $X$ be a smooth proper geometrically integral scheme over $\overline{\mathbb F_p}$. Assume $X$ is the specialization of a smooth proper scheme over $\mathbb Z_p^{nr}$. Let $L$ be an ample line ...

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158 views

### How close is $h^0(mD)$ to be a polynomial?

Let $X$ be a normal (or smooth if it helps) projective variety over an algebraically closed field $k$. Fix a Cartier divisor $D$: I am interested in knowing how $h^0(mD)$ behaves as $m$ varies.
At ...

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### Moving curves and small transformations

Let $f:X\dashrightarrow Y$ be an isomorphism in codimension one between smooth projective varieties. Let $C\subset X$ a curve generating an extremal ray of the cone of moving curves $Mov_1(X)$, and ...

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### Cone of moving curves

Let $X$ be a projective variety and $C\subset X$ be a moving curve, that is the curves numerically equivalent to $C$ cover a dense open subset of $X$.
How can we detect when $C$ is an extremal ray ...

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212 views

### Fibers of pushforward of a bundle when the fiber dimension is not constant

I could not decide if I should post this question in MO or Mathstackexchange, so feel free to downvote it if you think it does not belong here. I will delete my post and post it in MathSE in that case....

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**1**answer

137 views

### Big divisors in family

Given a family of divisors $D_t$ on varieties $X_t$, there are examples that show that bigness is not well behaved (e.g. example 2.2.13 in Positivity 1, shows we can have a special fiber where $D_0$ ...

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### Top intersections on the Hilbert scheme of points on a surface

The Picard group of $S^{[n]}$ is generated by the Picard group of $S$ (via a map $L \mapsto L_n$) and $E$, where $E = -\frac{B}{2}$, where $B$ is the exceptional divisor of the Hilbert Chow morphism.
...

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**1**answer

359 views

### Pull-back of the canonical divisor via a rational map

Let $f:X\dashrightarrow Y$ be a birational map between projective varieties not contracting any divisor. Assume that $X$ is smooth, and that $Y$ has at most ordinary singularities at finitely many ...

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**1**answer

190 views

### Number of conditions imposed by fat points to a linear system

Let $|D|$ be the linear system of degree $d$ hypersurfaces in $\mathbb{P}^n$ having multiplicity at least $m$ at $s$ general points.
Then $|kD|$ is the linear system of degree $kd$ hypersurfaces in $...

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**1**answer

143 views

### Why vanish the integer m of an ample line bundle in the Kodaira embedding theorem?

I try to understand the following version of the Kodaira embedding theorem:
Let $X$ be a compact Kähler manifold. A line bundle $L$ is positiv if and only if it is ample.
I have a problem with the '...

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vote

**1**answer

81 views

### Hermitic connections on complex line bundles with imaginary curvature form

It is a simple fact that if $L \to B$ is a complex line bundle endowed with an Hermitian product and a compatible connection $\nabla$, then the curvature $F_\nabla$ is imaginary (and so are the local ...

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### Moment map of equivariant line bundles

I'm reading Szabo's `Equivariant Cohomology and Localization of Path Integrals'. I've stumbled upon an equation I can't make sense of, in the discussion about $G$-equivariant line bundles on ...

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**1**answer

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### What is known about the cohomology of the relative tangent bundle on a conic bundle?

Let us assume $X$ is a smooth, projective and unirational variety of dimension $n$ over $\mathbb{C}$.
Given a conic bundle $\pi: Y\rightarrow X$ such that $\omega_{\pi}^{-1}$ is relatively very ample ...

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**1**answer

248 views

### Rigid effective divisors

Let $D\subset X$ be an effective smooth divisor in a smooth projective variety $X$. Assume that $h^0(X,D)=1$. In particular $D$ spans an extremal ray of the effective cone of $X$.
Now, let $f:X\...

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### How the existence of holomorphic sections depends on the choice of complex structure

In this Mathoverflow question it is asked how many invariant complex structures exist on the full flag manifold of $SU(m)$. In this question it is asked when a line bundle over a flag manifold has ...

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**1**answer

196 views

### Pencils in very ample linear systems without curve in its base locus

If $L$ is a very ample line bundle over a smooth complex projective surface $X$ and $s_0, \dots, s_n$ is a basis of the global sections of $L$, is there some choice of $i,j$ such that the pencil ...

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320 views

### Pushforward of line bundle under “toric isogeny”

Let $(X,T)$ be a smooth complex toric variety of dimension $d$ with torus $T$ and toric boundary $D=X\setminus T$. Let $\phi : X\to X$ be a finite endomorphism of $X$ such that the restriction
$$\phi|...

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136 views

### Line bundles with vanishing cohomology on Calabi-Yau manifold

Suppose we have some line bundle $L(D)$ on Calabi-Yau threefold. Let's call this line bundle "rigid" if $H^0(X,L(D)) \simeq \mathbb{C}$ and $H^i(X,L(D))=0$ for $i=1,2,3$.
Is anything known about such ...

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**1**answer

438 views

### Picard groups, ample cones, and proper birational maps

Let $f:Y\to X$ be a proper birational map of normal varieties over an algebraically closed field which is an isomorphism over the regular locus.
Q1: Is it the case that the pullback $f^*\...